What is Slope-Intercept Form

The slope-intercept form of a line equation is a commonly used formula in algebra that provides an easy way to identify the slope and y-intercept of a line. This form is particularly useful for quickly sketching a line on a coordinate plane or understanding the line’s direction and starting point. The slope-intercept form is expressed as:

\( y = mx + b \)

In this formula:

  • \( m \) is the slope of the line, which represents its steepness
  • \( b \) is the y-intercept, which is the point where the line crosses the y-axis

Derivation of the Slope-Intercept Form

To derive the slope-intercept form, we start with the concept of slope, defined as the “rise over run” between two points on a line. If we know one point on the line \( (x_1, y_1) \) and the slope \( m \), we can derive the equation of the line as follows:

  1. The slope \( m \) is defined as \( m = \dfrac{y – y_1}{x – x_1} \), where \( (x, y) \) is any other point on the line.
  2. Rearrange to solve for \( y \): \( y – y_1 = m(x – x_1) \).
  3. This equation, \( y – y_1 = m(x – x_1) \), is known as the point-slope form.
  4. Now, let’s assume that the line crosses the y-axis at \( (0, b) \), where \( b \) is the y-intercept.
  5. Substituting \( (x_1, y_1) = (0, b) \) into the point-slope form, we get \( y – b = m(x – 0) \), or \( y = mx + b \).

This is the slope-intercept form of the equation of a line.

How to Use the Slope-Intercept Form

The slope-intercept form, \( y = mx + b \), is valuable for quickly identifying key properties of a line:

  • Slope \( m \): Indicates how steep the line is. A positive slope means the line rises from left to right, while a negative slope means it falls.
  • Y-Intercept \( b \): The point where the line crosses the y-axis, or the starting value of \( y \) when \( x = 0 \).

Example 1: Writing an Equation in Slope-Intercept Form

Problem: Write the equation of a line with a slope \( m = 4 \) and y-intercept \( b = -3 \).

Solution:

  1. Use the slope-intercept form \( y = mx + b \).
  2. Substitute \( m = 4 \) and \( b = -3 \): \( y = 4x – 3 \).

The equation of the line is \( y = 4x – 3 \).

Example 2: Converting a Line Equation to Slope-Intercept Form

Problem: Convert the equation \( 3x – 2y = 6 \) to slope-intercept form.

Solution:

  1. Start with the equation \( 3x – 2y = 6 \).
  2. Isolate \( y \) by moving \( 3x \) to the other side: \( -2y = -3x + 6 \).
  3. Divide by \(-2\) to solve for \( y \): \( y = \dfrac{3}{2}x – 3 \).

The equation of the line in slope-intercept form is \( y = \dfrac{3}{2}x – 3 \).

The slope-intercept form \( y = mx + b \) is a powerful tool for analyzing and graphing lines, allowing us to quickly determine the slope and y-intercept of the line.